Spatial coherence and the persistence of high diversity in spatially heterogeneous landscapes

Abstract Our planet hosts a variety of highly diverse ecosystems. The persistence of high diversity is generally attributed to factors such as the structure of interactions among species and the dispersal of species in metacommunities. Here, we show that large contiguous landscapes—that are characterized by high dispersal—facilitate high species richness due to the spatial heterogeneity in interspecies interactions. We base our analysis on metacommunities under high dispersal where species densities become equal across habitats (spatially coherent). We find that the spatially coherent metacommunity can be represented by an effective species interaction‐web that has a significantly lower complexity than the constituent habitats. Our framework also explains how spatial heterogeneity eliminates differences in the effective interaction‐web, providing a basis for deviations from the area‐heterogeneity tradeoff. These results highlight the often‐overlooked case of high dispersal where spatial coherence provides a novel mechanism for supporting high diversity in large heterogeneous landscapes.

In the high dispersal limit, the effective dynamics is explained by the averaged interaction 19 matrix. Here we sketch a simple proof of this in one spatial dimension. We begin by arguing 20 that this result should hold even when the number of species and patches is low. 21 Consider two species over two patches. Species can only interact locally within each 22 patch, but they can disperse between patches. The system of equations is: For simplicity, we set r and K equal to 1. The fixed point equations for species 1 are: 24 φ 1,1 (1 − φ 1,1 ) + A 12,1 φ 1,1 φ 2,1 + D (φ 1,2 − φ 1,1 ) = 0 (S3) and 25 φ 1,2 (1 − φ 1,2 ) + A 12,2 φ 1,2 φ 2,2 + D (φ 1,1 − φ 1,2 ) = 0 (S4) Now note that adding up the fixed point equations for one species always gets rid of 26 the dispersal terms, irrespective of how many species or patches are present in the system.

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This results in the simple observation that the magnitude of D determines how close φ 1,2 32 and φ 1,1 should be. A sufficiently large D drives the densities at the two patches very 33

S2 SPATIAL COHERENCE AND THE AVERAGED INTERACTION MATRIX
close to one other, such that the densities are identical up to a few decimal places. The 34 equations for species 2 result in the same inference. Then, using φ 1,1 ∼ φ 1,2 and φ 2,1 ∼ φ 2,2 35 in equation S5: which is an effective fixed point equation for species 1 with an averaged interaction 37 strength A 12,1 +A 12,2 2 . A similar equation holds for species 2. The same procedure can 38 be extended for 2 patches containing any number of species. Note that the equilibrium 39 densities are not exactly equal at every patch, but rather they are same up to many decimal 40 places depending on how high the dispersal rate is.

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It is possible to generalize this procedure to any number of patches in 1 spatial dimension.

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We write down the fixed point equations for species 1 in the following manner: where f 1,α denotes the reaction term for species 1 at patch α. One can then write down 44 the following equation for every pair of connected patches such that indices α and α − 1 45 appear in the dispersal term: is obtained by multiplying fixed point equations corresponding to f 1,α ,

S2 SPATIAL COHERENCE AND THE AVERAGED INTERACTION MATRIX
For a given species, the reaction terms f would therefore be a mix of positives and 50 negatives in sign across different patches. This would partly ensure that the quantity 51 2f 1,α +f 1,α+1 −2f 1,α−1 −f 1,α−2 is of the same order of magnitude as any of the f s. Therefore, 52 a large D would imply that φ 1,α−1 − φ α is very small so as to match the contribution from Note that we already have a factor of 5 with the dispersal 54 term, which would bring down the D required to obtain sufficiently coherent densities, in 55 contrast to the 2 patch case. A large f would just mean that higher D is needed for coherent 56 densities. Using similar arguments, we can argue for spatially coherent densities for every 57 species.

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Since each of the f is of the form φ 1,α (1 − φ 1,α ) + φ 1,α j =1 A 1j,α φ j,α , therefore for 59 coherent densities equation S9 implies the effective fixed point equation: where N is the total number of patches. This demonstrates the average interaction 61 matrix result in the limit of spatial coherence.

S3 THE CASE OF VARYING CARRYING CAPACITIES AND DISPERSAL RATES
S3 The case of varying carrying capacities and dispersal rates 63 The averaged interaction matrix provides a good approximation of the equilibrium densi-64 ties, even when carrying capacities and dispersal rates are allowed to vary across species.

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Variance in carrying capacities can have a large effect on stability (Tang & Allesina, 2014). 66 We test whether the averaged interaction matrix still provides a good estimate of equilib-67 rium densities in the limit of high dispersal rates when carrying capacities are allowed to 68 vary (Fig. S1, S2). We consider local interaction strengths with sufficiently high standard 69 deviation, such that many species would go extinct in the absence of dispersal. If there is 70 no spatial correlation in interaction strengths, we find that very few species go extinct as 71 expected (Fig. S1). If the variance in carrying capacities is high, then spatial coherence 72 is achieved at higher dispersal rates that allow for accurate predictions via the averaged 73 interaction matrix. Averaged matrix species densities Figure S2: For the simulation settings described in Fig. S1, this plot shows the match between equilibrium species densities from patch 1 versus the corresponding estimates from the averaged interaction matrix.
We get similar results when dispersal rates are also allowed to vary across species, given 75 that the dispersal rates are sufficiently high (Fig. S3, S4). In Fig. S1 to S4, we show 76 comparisons with only one of the patches (patch 1), but the densities across all patches are 77 almost the same. and standard deviation for the average interaction matrix can be found as

S3 THE CASE OF VARYING CARRYING CAPACITIES AND DISPERSAL RATES
Here G * is a binomially distributed stochastic variable for how many of the entries X ij 89 are non-zero in the habitats.

S4 DERIVATION OF THE STANDARD DEVIATION OF THE EFFECTIVE INTERACTION MATRIX
The same procedure with entries in the habitat interaction matrices as stochastic vari-91 ables, X ijg , is followed in the correlated and nearest neighbour-correlated case. In the 92 correlated case however we do not need to treat the number of habitats G as a stochastic 93 variable (G * ), since the correlation ensures that an interaction is either zero or non-zero in 94 every habitat. For the correlated habitats we get where ρ nn denotes the nearest neighbour correlation.
diversity of the metacommunity. To test this three types of network topologies (in addition 105 to the grid in the main text), small world, random and Barabasi-Albert, were tested. In 106 these cases we used a generalisation of the discrete laplace operator according to where α is a habitat index and {α ← −} denote habitats connected to α. N {α− →} is the 108 degree of habitat α (number of habitats it is connected to). The h is the distance between 109 habitats, here set to 1.

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There were crucial differences between network topologies when diffusion rates were 111 low. We also found that a higher diffusion rate was needed to make the system spatially 112 coherent. Although, when spatial coherence was reached no differences were found between 113 the different network topologies and the the grid implementation.

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This tells us that our representation of the effective interaction matrix is valid in the 115 coherent limit regardless of the underlying habitat topology. On the other hand it also 116 suggests that it might be unreasonable to assume coherence for some topologies, because 117 of the high diffusion/dispersal rates needed. But luckily spatial coherence can be "easily"

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assessed, since the abundances should be approximately equal in all habitats involved.